Biological unit identification based on supervised shape ranking

ABSTRACT

A method of segmenting a digital image of biological tissue includes accessing a ranking model calculated from training data representing shapes of conforming and non-conforming biological unit exemplars. The ranking model may include support vectors defining a hyperplane in a vector space. The method further includes accessing image data representing the digital image, identifying a first shape and a set of second constituent shapes in the digital image, wherein the first shape comprises a union of the set of second constituent shapes, determining a rank of a first data point in the image data corresponding to the first shape and a rank of a second data point in the image data corresponding to the set of second constituent shapes into the vector space, and segmenting the digital image using the first shape or the set of second constituent shapes based on which data point has a greater respective rank.

FIELD

Embodiments relate generally to analysis of digital images, and moreparticularly, to analysis of digital images of biological tissuesamples.

BACKGROUND

The term segmentation, as used herein, refers to the identification ofboundaries of biological units, such as cells, within a digital image.The digital image may be obtained using a microscope. Weak or datadriven segmentations may be used to define cell boundaries. For example,a watershed transform is one image processing technique that has beenused for segmenting images of cells. With the watershed transform, adigital image may be modeled as a three-dimensional topological surface,where values of pixels (e.g., brightness or grey level) in the imagerepresent geographical heights.

Due to variations in the histology of different tissue types, however,weak segmentations may not produce an accurate segmentation withoutsignificant adaptation and optimization to specific tissue typeapplications. For example, a weak segmentation algorithm may cause theimage to be over-segmented (e.g., what appears as a single cell mayactually be only a portion of a cell) or under-segmented (e.g., whatappears as a single cell may actually be several different cells incombination). Furthermore, the image may not be properly segmented witha weak segmentation algorithm, in part, because a suitable segmentationparameter for one region of the image may not work well in other regionsof the same image. Therefore, a weak segmentation algorithm may not berobust enough for segmentation of large numbers of cells having manymorphological variations.

SUMMARY

One embodiment is directed to a computer-implemented method ofsegmenting a digital image of biological tissue. The computer includes aprocessor and a memory operatively coupled to the processor. The methodincludes accessing, in the memory, a plurality of support vectorscalculated from training data representing shapes of conformingbiological unit exemplars and shapes of non-conforming biological unitexemplars, the plurality of support vectors defining a hyperplane in avector space. The method further includes accessing, in the memory,image data representing the digital image of biological tissue,identifying, by the processor, a first shape and a set of secondconstituent shapes in the digital image using the image data, mapping,by the processor, a first data point in the image data corresponding tothe first shape and a second data point in the image data correspondingto the set of second constituent shapes into the vector space, andsegmenting, by the processor, the digital image using the first shape orthe set of second constituent shapes based on which of the first datapoint and the second data point has a greater respective signed distancefrom the hyperplane. The first shape comprises a union of the set ofsecond constituent shapes. In some embodiments, each segment of thedigital image may represent a cell.

In some embodiments, the method may include displaying, on a displayoperatively coupled to the processor, the segmented digital image usinga color coding of each cell, where the color coding represents a qualityof the segmentation.

In some embodiments, the digital image may be segmented using the firstshape where the signed distance between the hyperplane and the firstdata point is greater than the signed distance between the hyperplaneand the second data point. In some embodiments, the digital image may besegmented using the set of second constituent shapes where the signeddistance between the hyperplane and the second data point is greaterthan the signed distance between the hyperplane and the first datapoint.

In some embodiments, the method may include an act of storing thetraining data in the memory. In some embodiments, the method may includecomputing the plurality of support vectors using the processor. In someembodiments, the method may include computing, by the processor, alinear combination of each shape in the set of second constituentshapes. The second data point may correspond to the linear combination.The linear combination may, but need not, be applied to the kerneltransformation related to the support vector machine.

In some embodiments, the first shape and the linear combination of eachshape in the set of second constituent shapes may each be represented inthe image data as a histogram of points corresponding to a boundary ofthe first shape and linear combination, respectively, each point beinglocated on a polar coordinate plane. The method may include computing,by the processor, the first data point and the second data point usingthe histogram corresponding to the first shape and the linearcombination, respectively. In some embodiments, the method may includerotating each of the first shape and the linear combination of the setof second constituent shapes such that an axis of least inertia of therespective shape coincides with a zero degree radial of the polarcoordinate plane prior to computing the first data point and the seconddata point. The axis of least inertia may include a line from which theintegral of the square of distances to each point on the boundary of therespective shape is a minimum.

In some embodiments, the method may include applying, by the processor,a weak segmentation algorithm to the image data with certain parameters,for example, a watershed transform at a predetermined flooding level.

In some embodiments, the method may include identifying, by theprocessor, a set of third constituent shapes in the digital image usingthe image data. At least one shape in the set of second constituentshapes may comprise a union of the set of third constituent shapes. Themethod may further include mapping, by the processor, a third data pointcorresponding to the set of third constituent shapes into the vectorspace. The act of segmenting may include segmenting the digital imageusing the first shape, the set of second constituent shapes or the setof third constituent shapes based which of the first data point, thesecond data point and the third data point has a greater respectivesigned distance from the hyperplane.

In some embodiments, the digital image may be segmented using the firstshape where the signed distance between the hyperplane and the firstdata point is greater than either the signed distance between thehyperplane and the second data point and the signed distance between thehyperplane and the third data point. The digital image may be segmentedusing the set of second constituent shapes where the where the signeddistance between the hyperplane and the second data point is greaterthan either the signed distance between the hyperplane and the firstdata point and the signed distance between the hyperplane and the thirddata point. The digital image may be segmented using the set of thirdconstituent shapes where the signed distance between the hyperplane andthe third data point is greater than either the signed distance betweenthe hyperplane and the first data point and the signed distance betweenthe hyperplane and the second data point.

In some embodiments, the method may include applying, by the processor,a weak segmentation algorithm to the image data with certain parameters,for example, a watershed transform at a predetermined flooding level.

In one embodiment, a non-transitory computer-readable medium has storedthereon computer-executable instructions that when executed by acomputer cause the computer to access a plurality of support vectorscalculated from training data representing shapes of conformingbiological unit exemplars and shapes of non-conforming biological unitexemplars, access image data representing a digital image of biologicaltissue, identify a first shape and a set of second constituent shapes inthe digital image using the image data, wherein the first shapecomprises a union of the set of second constituent shapes, map a firstdata point in the image data corresponding to the first shape and asecond data point in the image data corresponding to the set of secondconstituent shapes into the vector space, and segment the digital imageusing one of the first shape and the set of second constituent shapesbased on which of the first data point and the second data point has agreater respective signed distance from a hyperplane. The plurality ofsupport vectors define the hyperplane in a vector space.

In some embodiments, the non-transitory computer-readable medium mayinclude computer-executable instructions that when executed by thecomputer cause the computer to apply a weak segmentation algorithm tothe image data at a predetermined flooding level to produce the set ofsecond constituent shapes.

In one embodiment, a system for segmenting a digital image of biologicaltissue includes a processor, an input coupled to the processor andconfigured to receive image data representing the digital image ofbiological tissue, and a memory coupled to the processor. The memoryincludes computer-executable instructions that when executed by theprocessor cause the processor to access a plurality of support vectorsbased on training data representing shapes of conforming biological unitexemplars and shapes of non-conforming biological unit exemplars,identify a first shape and a set of second constituent shapes in thedigital image using the image data representing the digital image ofbiological tissue, wherein the first shape comprises a union of the setof second constituent shapes, map a first data point in the image datacorresponding to the first shape and a second data point in the imagedata corresponding to the set of second constituent shapes into thevector space, and segment the digital image using one of the first shapeand the set of second constituent shapes based on which of the firstdata point and the second data point has a greater respective signeddistance from a hyperplane. The plurality of support vectors defines thehyperplane in a vector space.

In some embodiments, the digital image may be segmented using the firstshape where the where the signed distance between the hyperplane and thefirst data point is greater than the signed distance between thehyperplane and the second data point. In some embodiments, the digitalimage may be segmented using the set of second constituent shapes wherethe signed distance between the hyperplane and the second data point isgreater than the signed distance between the hyperplane and the firstdata point.

In some embodiments, the memory may include computer-executableinstructions that when executed by the processor cause the processor toapply a weak segmentation algorithm to the image data with certainparameters, for example, a watershed transform at a predeterminedflooding level, to produce the set of second constituent shapes. In someembodiments, the memory may further include computer-executableinstructions that when executed by the processor cause the processor tocompute a linear combination of each shape in the set of secondconstituent shapes, wherein the second data point corresponds to thelinear combination.

In some embodiments, the first shape and the linear combination of eachshape in the set of second constituent shapes may each be represented inthe image data as, but not limited to, a histogram of pointscorresponding to a boundary of the first shape and linear combination,respectively, each point being located on a polar coordinate plane. Thehistogram may contain appearance information besides the shapeinformation. Appearance information may include texture andintensity-based measurements. The memory may include computer-executableinstructions that when executed by the processor cause the processor tocompute the first data point and the second data point using thehistogram corresponding to the first shape and the linear combination,respectively. In some embodiments, the memory may includecomputer-executable instructions that when executed by the processorcause the processor to rotate each of the first shape and the linearcombination of the set of second constituent shapes such that an axis ofleast inertia of the respective shape coincides with a zero degreeradial of the polar coordinate plane prior to computing the first datapoint and the second data point, the axis of least inertia including aline from which the integral of the square of distances to each point onthe boundary of the respective shape is a minimum.

BRIEF DESCRIPTION OF THE DRAWINGS

Features and aspects of embodiments are described below with referenceto the accompanying drawings, in which elements are not necessarilydepicted to scale.

FIG. 1 depicts one example of a method of segmenting a digital image ofbiological tissue, in accordance with one embodiment.

FIG. 2A depicts one example of a class of biological unit exemplars in atraining set, in accordance with one embodiment.

FIG. 2B depicts one example of a class of non-biological unit exemplarsin the training set, in accordance with one embodiment.

FIG. 3 depicts one example of a digital image of cells segmented at onesegmentation scale level, in accordance with one embodiment.

FIG. 4A depicts one example of a digital image of a cell as detectedusing a shape descriptor, in accordance with one embodiment.

FIG. 4B depicts the cell of FIG. 4A oriented with respect to a polarcoordinate plane, in accordance with one embodiment.

FIG. 4C depicts a histogram of boundary points of the cell depicted inFIG. 3B, in accordance with one embodiment.

FIG. 5A depicts a comparison between one example of a cell segment andthe mean of several cell segments, in accordance with one embodiment.

FIG. 5B depicts one example of a mapping of the cell segments of FIG. 5Ainto a vector space, in accordance with one embodiment.

FIG. 6A depicts one example of a five-level hierarchical segmentation ofa portion of a digital image of cells, in accordance with oneembodiment.

FIG. 6B depicts examples of shape similarity trees for some of the cellsdepicted in FIG. 6A, in accordance with one embodiment.

FIG. 6C depicts one example of a mapping of some of the cells depictedin FIG. 6A into a vector space, in accordance with one embodiment.

FIGS. 7A-7M depict several examples of segmented digital images ofcells, in accordance with one embodiment.

FIG. 8 is a flow chart of one example of a method of segmenting adigital image, in accordance with one embodiment.

FIG. 9 is a block diagram of one example of a system for carrying outone or more embodiments.

DETAILED DESCRIPTION

Embodiments are directed to systems and methods of segmenting biologicalunits in a digital image of biological tissue. The digital image may beobtained, for example, using a microscope and a camera. The termbiological unit, as used herein, refers to discrete biologicalstructures and portions, components or combinations of biologicalstructures in the digital image. The target biological units in thedigital image may include, but are not necessarily limited to, cells.Exemplary target cells may include, for example, i) epithelial cellsand/or stromal cells, or ii) necrotic cells.

Histology patterns vary depending the type of biological tissue beingexamined. Therefore, it may be necessary to adapt and/or optimizesegmentation algorithms for specific tissue types. FIG. 1 depicts aschematic diagram of one example of a method of segmenting a digitalimage of biological tissue, according to one embodiment. The exemplarymethod provides an adaptive segmentation process that is based on aranking of the biological units in the digital image.

Given a training set of biological units, a user may rank eachbiological unit in the training set to generate a set of userannotations 10. For example, the user may rank the shape of eachbiological unit in the training set as “best,” “medium,” or “worst” interms of the quality of shape of the biological unit, or “conforming” or“non-conforming” based on the quality of the shape. The user annotations10 may then be used to learn a classifier or ranking model 20. Theclassifiers in the ranking model 20 may include, for example, RandomForest, Markov Random Field, Bayesian Networks, Belief Propagation,Support Vector Machines, Structural Support Vector Machines, and/orKernel Methods. The ranking model 20 may be associated with a function30 describing the energy or cost relationships between differentlyclassified biological units. These relationships may, for example,provide a quality measure to be used as a basis of comparison withbiological units in the target sample in the digital image.

One or more local operations 40 are defined by a set of rules that maybe applied to each biological unit identified at a particular scalelevel of an initially weakly segmented image 50. The local operations 40may include, for example, split, merge or erase operations performed ona hierarchical set of segmentations, such as described below withrespect to FIGS. 6A-6C, or using a library of predefined segmentations.The segmentation of the digital image may be optimized by accepting thelocal operations 40 having the lowest energy level or cost, according tothe function 30, that increase the rank of the biological unitsidentified in the proposed segmentation. Biological units in theproposed segmentation having a higher rank than biological units indifferent segmentations of the hierarchy have shapes that are mostsimilar to the shapes of the “best” or “conforming” biological units inthe training data. Local operations 40 that decrease the rank may berejected.

In some embodiments, a method of segmenting a digital image ofbiological tissue includes accessing a plurality of support vectorscalculated from training data. Support vectors are used by supportvector machines for classifying and analyzing data. A support vectormachine is a concept in statistics and computer science for analyzingdata and recognizing patterns with supervised learning methods. Such asupport vector machine may be used to predict for each input which ofone, two or more possible classes the input falls into (i.e., anon-probabilistic binary or multi-class classifier) based on trainingexemplars of the one, two or more classes. In one embodiment, thetraining data represents various shapes in each of one, two or moreclasses of exemplary shapes. FIG. 2A depicts a portion of a class ofexemplars 100 representing shapes that are cell-like (i.e., conformingshapes) and represent good segmentation. FIG. 2B depicts a portion of aclass of exemplars 102 representing shapes that are un-cell-like (i.e.,non-conforming shapes) and represent poor segmentation. The supportvectors represent exemplars in both classes (e.g., conforming andnon-conforming) of the training data mapped into a vector space, and maybe used to define a hyperplane in the vector space. The hyperplane is aline or curve separating one portion of the vector space containing oneclass of support vectors from another portion of the vector spacecontaining another class of support vectors for classifying the good andbad segmentation.

The method further includes identifying a first shape and a set ofsecond constituent shapes in the digital image. The shapes representcells segmented at different scale levels using a weak segmentationalgorithm (such as a watershed transform) for constructing ahierarchical segmentation of the digital image. For example, the firstshape may comprise a union of the set of second constituent shapes. Afirst data point representing the first shape is mapped into the vectorspace, and a second data point representing the set of secondconstituent shapes (e.g., a mean of the shapes in the set) is alsomapped into the vector space. The signed distance between the first datapoint and the hyperplane is compared to the signed distance between thesecond data point and the hyperplane. An optimum, or more optimal,segmentation of the digital image may be obtained using either the firstshape of the set of second constituent shapes based on whether the firstdata point or the second data point has a greater respective signeddistance from the hyperplane.

FIG. 3 depicts one example of cell segmentation of a two-dimensionalshape 200 at one scale level, according to one embodiment. Thetwo-dimensional shape 200 is represented as a closed and simple curve inEuclidean space. The interior and outermost boundary of the closed curve200 represents a single segment in the digital image at a higher scalelevel than depicted in FIG. 3. Depending on the accuracy of thesegmentation, the shape 200 may include a cell (i.e., the shape 200 isoptimally segmented), a portion of a cell (i.e., the shape 200 isover-segmented), or several cells in combination (i.e., the shape 200 isunder-segmented). As will be discussed below, the optimal segmentationof cells in the image may be obtained by comparing the similarity of thesegment shapes at two or more different segmentation scale levels to thetraining data. In this example, the shape 200 may be subdivided into aplurality of mutually-exclusive, non-overlapping constituent shapes 202a-202 j representing segmentation at a different scale level than thescale level of the singular shape 200. In other words, the shape 200 maybe expressed as the union of the constituent shapes 202 a-202 j.

In one embodiment, cell segmentation may be posed as an optimizationproblem where the cumulative cost of each shape in the image isminimized according to:

$\begin{matrix}{\mspace{79mu}{\min{\sum\limits_{i,l}\;{{Cost}\left( C_{i}^{l} \right)}}}\;} & (1)\end{matrix}$where C is the set of i curves (also referred to herein as shapes orcell segments) at scale level l.

Minimizing Eq. (1) is an NP-hard optimization problem, given that thenumber of different shape-to-shape combinations grows exponentially asthe number of scale levels and segments increase. To reduce complexity,Eq. (1) may be minimized by constraining the number of segmentationcombinations such that a subset of shapes at one segmentation scalelevel are contained in one and only one shape at the next adjacentsegmentation scale level, i.e., a shape at one scale level is the unionof a subset of shapes at an adjacent lower level. Referring to FIG. 3,for example, the shape 200 is the union of the constituent shapes 202a-202 j under this constraint.

In one embodiment, the boundaries or shapes and scales of cells may beidentified based on a predetermined shape descriptor. The shapedescriptor may be used to define a measure of similarity between shapesin the image and the exemplars in the training data.

The similarity in shape and scale between two cells in a digital imagemay be determined using a polar coordinate-based shape descriptor (e.g.,a two-dimensional ring, such as described above, located in a polarcoordinate plane). FIG. 4A depicts one example of a digital image of acell 300. Initially, as depicted in FIG. 4B, the cell 300 may be rotatedso that the zero degree radial 310 of the polar coordinate planecoincides with an orientation of the cell. The orientation of the cell,which is the axis of least inertia (ALI), may be defined as the line forwhich the integral of the square of the distances from the pole of thepolar coordinate plane to each point on the boundary of the cell 300 isa minimum. After the cell 300 is oriented, the boundary points of thecell may be sampled using a morphological operation. For example, asillustrated in FIG. 4B, the polar coordinate plane may be divided into12 30-degree bins. Each boundary point of the cell 300 may berepresented in the polar coordinate system by a two-tuple (θ, ρ), whereθ denotes the angular coordinate of the boundary point and ρ is thedistance between the pole and the boundary point. A histogram of allboundary points, such as depicted in FIG. 4C, may be generated byprojecting each point into a corresponding bin according to thefollowing formula:

$\begin{matrix}{{{{His}(i)} = {\frac{1}{\sum\limits_{\forall j}^{\;}\;{B(j)}}{\sum\limits_{\forall{\theta \in {B{(i)}}}}^{\;}\;{\log\left( {1 + \frac{1}{\rho}} \right)}}}},} & (2)\end{matrix}$where B(i) denotes the degree interval of the ith bin and

$\frac{1}{\sum\limits_{\forall j}^{\;}\;{B(j)}}$is a normalization term. In FIG. 4C, the horizontal axis represents eachof the 12 bins in the polar coordinate plane, and the vertical axisrepresents His(i) for each corresponding bin i. Because the shapedescriptor is translation, rotation and scale variant, the logarithmfunction makes the shape descriptor more sensitive to the boundarypoints near to the pole than those of points far away.

For a given training set of shapes having one, two or more classes(e.g., a class of positive exemplars and a class of negative exemplars),a distance metric d can be defined as the projection of a given pointwith respect to the hyperplane, which as discussed above, separates theclass of conforming support vectors from the class of non-conformingsupport vectors. Since the elements of the training dataset are linearseparable, the hyperplane can be expressed in terms of the supportvectors as:w=Σ _(i) ^(N) ^(s) α_(i) y _(i) x _(i)  (3)where N_(s) is the number of support vectors, y_(i) is the correspondingclass (e.g., conforming or non-conforming), α_(i) are the Lagrangemultipliers. The signed distance function d(x) from any point in thevector space to the hyperplane is:

$\begin{matrix}{\mspace{79mu}{{{d(x)} = {\frac{\left\langle {w,x} \right\rangle + b}{{w}^{2}} = \frac{{\sum\limits_{i}^{N_{s}}\;{\alpha_{i}y_{i}\left\langle {x_{i},x} \right\rangle}} + {b}}{{{\sum\limits_{i}^{N_{S}}{\alpha_{i}y_{i}x_{i}}}}^{2}}}};}} & (4)\end{matrix}$If d>0, then x is more similar to the conforming class, and if d<0, x ismore similar to the non-conforming class. If φ:R^(n)→H is a mapping fromR^(n) to the Hilbert space H and K is a kernel function defined as:(x_(i),x_(j))=<φ(x_(i)),φ(x_(j))>, then the signed distance functiond(x) can be rewritten as:

$\begin{matrix}{\mspace{79mu}{{d(x)} = \frac{{\sum\limits_{i}^{N_{s}}\;{\alpha_{i}y_{i}\left\langle {{\phi\left( x_{i} \right)},{\phi(x)}} \right\rangle}} + b}{{{\sum\limits_{i}^{N_{s}}\;{\alpha_{i}y_{i}{\phi\left( x_{i} \right)}}}}^{2}}}} & (5)\end{matrix}$

According to one embodiment, the function d(x) as in Eq. (5) induces aranking function that can be used to compare two shapes. For two shapesC_(A) and C_(B), C_(A) is more similar to the conforming class thanC_(B) if d(C_(A))>d(C_(B)). Based on this, a similarity tree can beconstructed. If C_(i) ^(l)=∪C_(j) ^(l-1),∩C_(j) ^(l-1)=∅ then a subsetof shapes at level l−1 in terms of a shape at level l as:d(C _(j) ^(l-1))=d(C _(j) ^(l))+d(Δ_(j) ^(l-1))  (6)

Eq. (6) can be rewritten as:

$\begin{matrix}{\mspace{79mu}{{{d\left( C_{j}^{l} \right)} = {{\frac{\sum{d\left( c_{j}^{l - 1} \right)}}{n} - \frac{\sum{d\left( \Delta_{j}^{l - 1} \right)}}{n}} = {{\mu\left( C_{j}^{l - 1} \right)} - {\mu\left( \Delta_{j}^{l - 1} \right)}}}},}} & (7)\end{matrix}$where

$\mspace{79mu}{{\mu\left( C_{j}^{l - 1} \right)} = {d\left( \frac{\sum\left( c_{j}^{l - 1} \right)}{n} \right)}}$     and$\mspace{79mu}{{\mu\left( \Delta_{j}^{l - 1} \right)} = {{d\left( \frac{\sum\left( \Delta_{j}^{l - 1} \right)}{n} \right)}.}}$The geometrical interpretation of

$\mspace{79mu}\frac{\sum}{n}$      and$\mspace{79mu}{d\left( \frac{\sum\left( \Delta_{j}^{l - 1} \right)}{n} \right)}$is the mean shape of the partition and the mean shape differences of thepartitions with respect to the single shape, respectively. A many-to-oneranking function may be expressed in terms of the signed distance d(x)as:d({C ₁ ^(l) , . . . , C _(j) ^(l)})<Δd(C _(i) ^(l))

μ(Δ_(j) ^(l-1))>0  (8)Eq. (8) provides one example of an efficient way to compute similaritiesbetween subgroups of shapes in the image.

According to one embodiment, FIG. 5A depicts a comparison between oneexample of a shape 400 and the mean shape 402 of shapes 400 a, 400 b and400 c. The shape 400 includes the union of shapes 400 a, 400 b and 400c, which represent segmentations at two consecutive scale levels,respectively. In other words, shapes 400 a, 400 b and 400 c represent asegmentation at scale level l−1, and the shape 400 represents adifferent segmentation at scale level l, which is hierarchically onelevel above l−1. FIG. 5B depicts the mapping of shapes 400 and 402 intoa vector space 410. A plurality of support vectors representingconforming shapes 412 and a plurality of support vectors representingnon-conforming shapes 414 may be used to define a hyperplane 416 thatseparates the vector space 410 into two regions (marked as “+” for theconforming region and “−” for the non-conforming region). Dashed lines418 and 419 represent margins about the hyperplane 416. A first datapoint 420 in the vector space 410 represents shape 400, which falls inthe conforming region of the vector space 410. A second data point 430in the vector space 410 represents the mean shape 402, which also fallsin the conforming region of the vector space 410. For perspective, eachof the shapes 400 a, 400 b and 400 c is depicted at points 422, 424 and426, respectively, in the vector space 410. A first signed distance 440is the signed distance between the first data point 420 and thehyperplane 416, and a second signed distance 442 is the signed distancebetween the second data point 430 and the hyperplane 416. As can beenseen in this example, the first signed distance 440 exceeds the secondsigned distance 442 because the first data point 420 is further from thehyperplane 416 in the conforming region than the second data point 430.

As discussed above with respect to FIG. 3, in some embodiments,topological constraints may be imposed on a weak segmentation atdifferent scale levels and with different paramters. FIG. 6A depicts oneexample of a five-level hierarchical segmentation of a portion of adigital image of biological tissue, according to an embodiment. It willbe appreciated that any number of levels and parameters of a weaksegmentation may be used. In this example, each level A through Erepresents a different level of segmentation, where level E is thehighest level in which there is only one shape and level A is the lowestlevel, which includes many constituent shapes. Intermediate levels B, Cand D each include various combinations of constituent shapes from theadjacent level. Topological constraints may be imposed for minimizingEq. (1) in which each shape at one segmentation scale level is includedin one and only one shape at an adjacent segmentation scale level. Forexample, in FIG. 6A, shape B₁, which is one shape at scale level B, isthe union of shapes A₁, A₂ and A₃, which are shapes at an adjacent(lower) scale level A. Furthermore, the intersection of shapes A₁, A₂and A₃ at scale level A with any shapes at scale level B other thanshape B₁ is an empty set. In this embodiment, these topologicalconstraints apply to all shape segments at all scale levels.

Using these constraints, the segmentation at various differenthierarchical scale levels (e.g., scale levels A, B, C, D and E) may berepresented as a shape similarity tree. FIG. 6B depicts examples ofshape similarity trees 520 and 530 for some of the shapes depicted inFIG. 6A, according to one embodiment. The root of similarity tree 520 isnode 521, which represents the shape with the minimum cost (in thisexample, shape C₁ indicated at 521), and the leaves of similarity tree520 are nodes 524-529 (in this example, shapes A₁-A₆ indicated at524-529). Shapes at a given segmentation scale level in the similaritytree are ranked with respect to shapes at adjacent segmentation scalelevels according to the ranking function d(x) (Eq. 8). For example, asdepicted in FIG. 6B, shape C₁ 521 may be ranked with respect toconstituent shapes B₁ 522 and B₂ 523 according to Eq. (8). Likewise,shapes B₁ 522 and ₂ 523 may be ranked with respect to constituent shapesA₁-A₃ (524-526) and A₄-A₆ (527-529), respectively. The shape or set ofshapes which most closely match the conforming exemplars (e.g., such asthe conforming exemplars 100 depicted in FIG. 2A) can be determined as afunction of the minimum cumulative cost of each shape at a plurality ofsegmentation scale levels in the image:

$\begin{matrix}{\mspace{79mu}{{\min{\sum\limits_{i,l}\;{{Cost}\left( C_{i}^{l} \right)}}} = {\max{\sum\limits_{i,l}\left( \frac{1}{1 + {d\left( c_{i}^{l} \right)}} \right)}}}} & (9)\end{matrix}$

In some embodiments, shape similarity trees can be constructed byrecursively merging constituent shapes at lower scales (e.g., at scalelevel l−1) to create a larger shape comprising the union of theconstituent shapes if the ranking of the larger shape is higher than theranking of a mean of the constituent shapes at the lower scale level.

FIG. 6C depicts one example of the mapping of shape segments of an imageinto a vector space 550. A plurality of support vectors representingconforming shapes 540 and a plurality of support vectors representingnon-conforming shapes 542 are used to define a hyperplane 544 thatseparates a vector space 550 into two regions (marked as “+” for theconforming region and “−” for the non-conforming region). Dashed lines546 and 548 represent margins about the hyperplane 544. A first datapoint 521 corresponding to shape C₁ is mapped into the conforming region(“+”) of the vector space 550. A second data point 560 representing amean of the shapes B₁ 522 and B₂ 523 is mapped into the vector space550, which also falls in the conforming region (“+”) of the vectorspace. A first signed distance 570 is the signed distance between thefirst data point 521 and the hyperplane 544, and a second signeddistance 572 is the signed distance between the second data point 560and the hyperplane 544. As can been seen in this example, the firstsigned distance 570 exceeds the second signed distance 572 because thefirst data point 521 is further from the hyperplane 544 in theconforming region than the second data point 560. Thus, according to Eq.(8), shape C₁ 521 is ranked higher than the mean of shapes B₁ 522 and B₂523.

This same principle applies to mappings of shapes A₁-A₃ (524-526) andA₄-A₆ (527-529), and their respective mean shapes 562 and 564. Since thesigned distances 573 and 574 between mean shapes 562 and 564,respectively, and the hyperplane 544 are less than the signed distancesbetween the mean shape 560 and the hyperplane 544, shapes B₁ and B₂,which include constituent shapes A₁-A₃ (524-526) and A₄-A₆ (527-529),respectively, are ranked higher than shapes A₁-A₃ (524-526) and A₄-A₆(527-529). These exemplary rankings are reflected in the shapesimilarity tree 520 of FIG. 6B. Therefore, effectively among all shapesA₁-A₆ (524-529), B₁-B₂ (522-523) and C₁ 521, shape C₁ is most similar tothe shape exemplars that conform to the biological units in the trainingdata (e.g., exemplars 100 of FIG. 2A) and thus represents the optimalsegmentation among all shapes in the similarity tree. Likewise, shapesA₁-A₆ (524-529) are most similar to the non-conforming shape exemplarsin the training data (e.g., exemplars 102 of FIG. 2B) and are thusrepresent the least optimal segmentation. Once the ranking of the shapesin the similarity tree are established, the optimal shapes can bedetermined using either a top-down or bottom-up analysis since eachtechnique converges to the same global cost minima (Eq. 9).

The segmentation ranking results may represent a quality metric. In someembodiments, the segmentation results may displayed with a color codingto represent the quality of the relevant portion of the segmentation.For example, in one embodiment, the cells in the target digital imagethat are most similar to the conforming exemplars in the training dataare color coded in a first color, e.g., green; those that are lesssimilar are color coded in a second color, e.g., yellow; and those leastsimilar to the conforming exemplars are color coded in a third color,e.g., red. Any number of colors related to any number of cell classes orcategories can be used for displaying the segmentation quality metric.Additionally or alternatively, the segmentation results may displayedwith a color intensity coding to represent the quality of the relevantportion of the segmentation. For example, in one embodiment, the cellsin the target digital image that are most similar to the conformingexemplars in the training data are color coded in a first intensity,e.g., very intense green; those least similar to the conformingexemplars are color coded in a second intensity, e.g., a green that isnot intense; and those that are somewhere between in similarity arecolor coded in an intermediate intensity. Such a display may enable thequality of the segmentation to be readily apparent to the user.

FIGS. 7A-7M depict several examples of digital images of cells prior toand following segmentation using techniques according to one or moreembodiments, as taught herein. To visualize the results, the segmentedcells may be color-coded according to the ranking function and/or thenumber of cell-classes. For example, in a two-class problem (e.g.,positive and negative cell classes), cells colored in green may have thehighest ranking and maximum shape similarity with respect to thepositive cell class, while cells colored in red may have the lowestranking and minimum shape similarity with respect to the positive cellclass and maximum similarity with respect to the negative cell class),and cells colored in yellow may have intermediate rankings and shapesimilarities. In FIGS. 7A-7M, the highest ranking cells are depicted asunfilled shapes with solid outlines, the lowest ranking cells aredepicted with horizontal hatch marks, and intermediately ranked cellsare depicted with diagonal hatch marks. FIG. 7A shows an originalepithelial cell image, while FIG. 7B shows an overlay of the epithelialcell image with the segmentation results after applying one segmentationtechnique as taught herein. FIG. 7C shows an image with cell nuclei.

FIG. 7D shows the nuclei segmentation result after applying onesegmentation technique as taught herein. FIGS. 7E-7H show comparativeresults of applying a morphological watershed algorithm at threedifferent levels: high, medium and low, and a segmentation afterapplying one segmentation technique as taught herein. Note the over andunder segmentation effects in the first three cases as oppose to theresults obtained using a segmentation technique as taught herein. FIGS.7I-7K show details of epithelial cells with low, medium and high rankingrespectively. Note that the cell in FIG. 7I has an irregular shape andis relatively larger than others, while the cell in FIG. 7J has anelliptical-like shape, and cells in FIG. 7K have smooth contours andcircular-like shapes. FIGS. 7L and 7M show details of the segmentationresults as applied to nuclei using a segmentation technique as taughtherein. Note that cells with the high rank (which may be colored ingreen) have an elliptical and circular type of shape as compared withcells having a low rank (which may be colored in red). It will beunderstood that embodiments taught herein may be generalized and appliedto other types of biological tissue. Different models (i.e.,hyperplanes) may be created for different cell morphologies to enablethe segmentation techniques as taught herein to be applied to thosecells morphologies. Further, the method is not limited to a two-classcell problem as it can be used for solving a one-class or multi-classcell problem.

FIG. 7 is a flow diagram of one example of a process 700 of segmenting adigital image, according to one embodiment. Process 700 begins at block702. At block 704, a plurality of support vectors are accessed (e.g.,from a memory of a computer). The support vectors may be pre-computedfrom training data representing a set of conforming exemplary cells anda set of non-conforming exemplary cells. Further, the support vectorsmay be used to define a hyperplane in a vector space, where thehyperplane separates support vectors representing the conforming cellsfrom support vectors representing non-conforming cells, such as depictedin the example of FIG. 6C.

At block 706, image data representing a target digital image is accessed(e.g., from the memory). The digital image may include an image ofbiological tissue (including cells). At block 708, at least one firstshape is identified in the image using a conventional segmentationtechnique, such as, but not limited to, a watershed transform, meanshift segmentation, graph based segmentation and/or normalized cuts. Forexample, referring to FIG. 6A, the first shape may be shape B₁. Furtherat block 708, a set of second constituent shapes are identified in theimage. In this example, the set of second constituent shapes are shapesthat constitute shape B₁ at the adjacent scale level, and include shapesA₁, A₂ and A₃. As can be seen, shape B₁ is the union of shapes A₁, A₂and A₃, and none of the latter shapes intersects any hierarchical shapeother than shape B₁. In this example, only shapes at two adjacent scalelevels are used; however, it will be appreciated that shapes at morethan two scale levels may be used by iteratively identifying shapes atdifferent scale levels. The boundaries of each shape may be identifiedin the segmented image using, for example, a multiscale analysis basedon a shape descriptor (e.g., a two-dimensional variable diameter ring)as in Eq. (2), described above.

At block 710, a first data point representing the first shape (e.g.,shape B₁) is mapped into the vector space. Referring to the example ofFIG. 6C, the first data point may be data point 523. Further, a seconddata point representing the set of second constituent shapes (e.g.,shapes A₁, A₂ and A₃) is mapped into the vector space. Referring againto the example of FIG. 6C, the second data point may data point 562,which represents the mean of shapes A₁, A₂ and A₃. At block 712, thesigned distance between the first data point (e.g., point 523) and thehyperplane (e.g., hyperplane 544) is compared to the signed distancebetween the second data point (e.g., point 562) and the hyperplane. Ifthe signed distance of the first data point is greater than the signeddistance of the second data point, process 700 proceeds to block 714;otherwise, process 700 proceeds to block 716. In the example of FIG. 6C,since the signed distance of the first data point is positive and thesigned distance of the second data point is negative, process 700 willproceed to block 714.

At block 714, the digital image is segmented using the first shape(e.g., shape B₁), since it was determined above that the first shape ismore similar to the conforming exemplars than the set of secondconstituent shapes. At block 716, the digital image is segmented usingthe set of second constituent shapes instead of the first shape, sinceit was determined above that the set of second constituent shapes ismore similar or no more similar to the conforming exemplars than thefirst shape. It should be noted that the above-described process may beiteratively repeated for first shapes and sets of second constituentshapes at different adjacent scale levels (e.g., levels A and B, B andC, C and D, and D and E) to identify the optimum segmentation for thedigital image. As discussed above, the optimum segmentation may bedetermined using either a top-down (highest scale to lowest scale) orbottom-up (lowest scale to highest scale) analysis. From either block714 or 716, process 700 proceeds to end at block 718.

FIG. 9 is a block diagram of one example of a system 810 in accordancewith one or more embodiments. In FIG. 9, system 810 includes a processor814 and a display 816. Memory 812 may include any suitable memoryassociated with the processor, such as ROM (read only memory), RAM(random access memory) or DRAM (dynamic random access memory), or anysuitable non-transitory memory medium, such as a DVD, CD or memory card.In some embodiments, processor 814 includes memory 812 and/or display816. In these embodiments, memory 812 includes executable instructionsfor performing one or more of the methods or processes taught herein. Inother embodiments, the memory 812 and/or display 816 may becommunicatively coupled to the processor 814, but not part of processor814. In these embodiments, the memory 812 and/or display 816 may stillbe accessed through any suitable connection device or communicationsnetwork including but not limited to local area networks, cablenetworks, satellite networks, and the Internet, regardless whether hardwired or wireless. In some embodiments, memory 812 is both included inprocessor 814 and communicatively coupled to processor 814. One or moreelements of memory 812 may further include image data representing atleast one digital image of biological tissue. One or more elements ofmemory 812 may be configured to store data representing a plurality ofsupport vectors that define a hyperplane in vector space and that havebeen calculated from training data. One or more elements of memory 812may be configured to store training data representing shapes ofconforming biological unit exemplars and shapes of non-conformingbiological unit exemplars. For example, the training data may includeimages of biological tissue in which cells have been segmented andclassified as either good segmentations or poor segmentations. Theprocessor 814, or CPU, may comprise a microprocessor, microcontrollerand a digital signal processor (DSP).

In some embodiments, the processor 814 may be configured to access theimage data, the training data and/or the support vectors stored in thememory 812 or in a memory of a remote device 826.

The memory 812 and the processor 814 may be incorporated as componentsof an analytical device such as an automated high-speed system thatimages and analyzes in one system. Examples of such systems include, butare not limited to, General Electric's InCell analyzing systems (GeneralElectric Healthcare Bio-Sciences Group, Piscataway, N.J.). In some suchembodiments, system 810 may include a digital imager 830, an interactiveviewer 818, and a virtual microscope 820. Digital imager 830 may be, forexample, a fluorescent imaging microscope having an excitation source832 and configured to capture digital images of the biological samples.In embodiments that are part of a larger analytical device, system 810may include a network interface 822 for transmitting one or more of theimages or any related data or analytical information over acommunications network 824 to one or more remote systems 826.

The network interface 822 in embodiments of system 810 may include anycomponents configured to transmit and/or receive data over acommunications network, including hardwired or wireless digitalcommunications systems. In embodiments in which system 810 includesnetwork interface 822, network interface 822 may additionally oralternatively be used for receiving one or more of the images or anyrelated data or analytical information over a communications network 824from one or more remote systems 826.

System 810 and/or system 826 may include a display 816. The display 816may include any device capable of displaying a digital image, such asdevices that incorporate an LCD or CRT.

In some embodiments, the memory 812 may include executable code forperforming segmentation of cells or other biological units, includingcalculating the support vectors from the training data, identifying thefirst shape and the set of second constituent shapes using the imagedata, mapping the first and second data points corresponding to thefirst shape and set of second constituent shapes, respectively, and/orsegmenting the digital image using the first shape or the set of secondconstituent shapes based on the respective distances of the first andsecond data points from the hyperplane. One of ordinary skill in the artwill understand that many known automated segmentation methods andtechniques may be employed in conjunction with the methods taughtherein, which may include watershed feature detection, statisticallydriven thresholding, (e.g., Otsu, mean, MinError, Huang, triangles, andMinMax thresholding) and/or edge enhancing filters (e.g., unsharpmasking, Sobel filtering, Gaussian filters, Kalman filters). In someembodiments, a conventional segmentation technique (e.g., a watershedtransform) may be implemented in the executable code for generating aweak segmentation for identifying the first shape and/or the set ofsecond constituent shapes. In some embodiments, the executable code mayinclude functionality for user-assisted segmentation of cells or otherobjects (e.g., tools allowing users to indicate cell or objectboundaries).

Embodiments taught herein may be used in a variety of applications, suchas cell differentiation, cell growth, cell movement and tracking, andcell cycle analysis. Cell differentiation includes identification ofsubpopulations of cells within cell clusters. Such information may beuseful in many different types of cellular assays, such as co-cultureassays in which two or more different kinds of cells are grown together.

Having thus described several exemplary embodiments of the invention, itis to be appreciated various alterations, modifications, andimprovements will readily occur to those skilled in the art. Forexample, in some embodiments, digital images of biological units otherthan cells may be segmented. The biological units may include biologicalstructures larger than cells, such as the lens of an eye, a heart valve,or an entire organ. Such alterations, modifications, and improvementsare intended to be part of this disclosure, and are intended to bewithin the scope of the invention. Accordingly, the foregoingdescription and drawings are by way of example only.

What is claimed is:
 1. A computer-implemented method of segmenting adigital image of biological tissue, the computer including a processorand a memory operatively coupled to the processor, the method comprisingacts of: accessing, in the memory, a ranking model calculated fromtraining data representing shapes of conforming biological unitexemplars and shapes of non-conforming biological unit exemplars, theranking model defining a ranking of a quality of the shapes in thetraining data with respect to the conforming biological unit exemplarsand the non-conforming biological unit exemplars; accessing, in thememory, image data representing the digital image of biological tissue;identifying, by the processor, a first shape and a set of secondconstituent shapes in the digital image using the image data, whereinthe first shape comprises a union of the set of second constituentshapes; determining, by the processor, a rank of a first data point inthe image data corresponding to the first shape and a rank of a seconddata point in the image data corresponding to the set of secondconstituent shapes using the ranking model; and segmenting, by theprocessor, the digital image using one of the first shape and the set ofsecond constituent shapes based on which of the first data point and thesecond data point has a greater respective rank.
 2. Thecomputer-implemented method of claim 1, wherein: the ranking modelincludes a plurality of support vectors calculated from the trainingdata, the plurality of support vectors defining a hyperplane in a vectorspace; the act of determining includes mapping, by the processor, thefirst data point and the second data point into the vector space; andthe act of segmenting includes segmenting, by the processor, the digitalimage using one of the first shape and the set of second constituentshapes based on which of the first data point and the second data pointhas a greater respective distance from the hyperplane.
 3. Thecomputer-implemented method of claim 2, wherein the digital image issegmented using the first shape where the signed distance between thehyperplane and the first data point is greater than the signed distancebetween the hyperplane and the second data point.
 4. Thecomputer-implemented method of claim 2, wherein the digital image issegmented using the set of second constituent shapes where the signeddistance between the hyperplane and the second data point is greaterthan the signed distance between the hyperplane and the first datapoint.
 5. The computer-implemented method of claim 2, further comprisingan act of storing the training data in the memory.
 6. Thecomputer-implemented method of claim 2, further comprising an act ofcomputing the plurality of support vectors using the processor.
 7. Thecomputer-implemented method of claim 2, further comprising an act ofcomputing, by the processor, a linear combination of each shape in theset of second constituent shapes, wherein the second data pointcorresponds to the linear combination.
 8. The computer-implementedmethod of claim 7: wherein the first shape and the linear combination ofeach shape in the set of second constituent shapes are each representedin the image data as a histogram of points corresponding to a boundaryof the first shape and linear combination, respectively, each pointbeing located on a polar coordinate plane, and wherein the methodfurther comprises an act of computing, by the processor, the first datapoint and the second data point using the histogram corresponding to thefirst shape and the linear combination, respectively.
 9. Thecomputer-implemented method of claim 8, further comprising an act ofrotating each of the first shape and the linear combination of the setof second constituent shapes such that an axis of least inertia of therespective shape coincides with a zero degree radial of the polarcoordinate plane prior to computing the first data point and the seconddata point, the axis of least inertia including a line from which theintegral of the square of distances to each point on the boundary of therespective shape is a minimum.
 10. The computer-implemented method ofclaim 2, further comprising an act of applying, by the processor, a weaksegmentation algorithm to the image data to produce the set of secondconstituent shapes.
 11. The computer-implemented method of claim 2,further comprising acts of: identifying, by the processor, a set ofthird constituent shapes in the digital image using the image data,wherein at least one shape in the set of second constituent shapescomprises a union of the set of third constituent shapes; and mapping,by the processor, a third data point corresponding to the set of thirdconstituent shapes into the vector space, wherein the act of segmentingincludes segmenting the digital image using one of the first shape, theset of second constituent shapes and the set of third constituent shapesbased which of the first data point, the second data point and the thirddata point has a greater respective signed distance from the hyperplane.12. The computer-implemented method of claim 11: wherein the digitalimage is segmented using the first shape where the signed distancebetween the hyperplane and the first data point is greater than eitherthe signed distance between the hyperplane and the second data point andthe signed distance between the hyperplane and the third data point,wherein the digital image is segmented using the set of secondconstituent shapes where the where the signed distance between thehyperplane and the second data point is greater than either the signeddistance between the hyperplane and the first data point and the signeddistance between the hyperplane and the third data point, and whereinthe digital image is segmented using the set of third constituent shapeswhere the signed distance between the hyperplane and the third datapoint is greater than either the signed distance between the hyperplaneand the first data point and the signed distance between the hyperplaneand the second data point.
 13. The computer-implemented method of claim11, further comprising an act of applying, by the processor, a weaksegmentation algorithm to the image data to produce the set of thirdconstituent shapes.
 14. The computer-implemented method of claim 2,wherein each segment of the digital image represents a cell.
 15. Thecomputer-implemented method of claim 14, further comprising displaying,on a display operatively coupled to the processor, the segmented digitalimage using a color coding of each cell, wherein the color codingrepresents a quality of the segmentation.
 16. A non-transitorycomputer-readable medium having stored thereon computer-executableinstructions that when executed by a computer cause the computer to:access a ranking model calculated from training data representing shapesof conforming biological unit exemplars and shapes of non-conformingbiological unit exemplars, the ranking model defining a ranking of aquality of the shapes in the training data with respect to theconforming biological unit exemplars and the non-conforming biologicalunit exemplars; access image data representing a digital image ofbiological tissue; identify a first shape and a set of secondconstituent shapes in the digital image using the image data, whereinthe first shape comprises a union of the set of second constituentshapes; determine a rank of a first data point in the image datacorresponding to the first shape and a rank of a second data point inthe image data corresponding to the set of second constituent shapesinto the vector space; and segment the digital image using one of thefirst shape and the set of second constituent shapes based on which ofthe first data point and the second data point has a greater respectiverank.
 17. The non-transitory computer-readable medium of claim 16,further comprising computer-executable instructions that when executedby the computer cause the computer to apply a weak segmentationalgorithm to the image data to produce the set of second constituentshapes.
 18. A system for segmenting a digital image of biological tissuecomprising: a processor; an input coupled to the processor andconfigured to receive image data representing the digital image ofbiological tissue; and a memory coupled to the processor, the memoryincluding computer-executable instructions that when executed by theprocessor cause the processor to: access a ranking model based ontraining data representing shapes of conforming biological unitexemplars and shapes of non-conforming biological unit exemplars, theranking model defining a ranking of a quality of the shapes in thetraining data with respect to the conforming biological unit exemplarsand the non-conforming biological unit exemplars; identify a first shapeand a set of second constituent shapes in the digital image using theimage data representing the digital image of biological tissue, whereinthe first shape comprises a union of the set of second constituentshapes; determine a rank of a first data point in the image datacorresponding to the first shape and a rank of a second data point inthe image data corresponding to the set of second constituent shapesinto the vector space; and segment the digital image using one of thefirst shape and the set of second constituent shapes based on which ofthe first data point and the second data point has a greater respectiverank.
 19. The system of claim 18: wherein the ranking model includes aplurality of support vectors calculated from the training data, theplurality of support vectors defining a hyperplane in a vector space,and wherein the memory further includes computer-executable instructionsthat when executed by the processor cause the processor to: map thefirst data point and the second data point into the vector space; andsegment the digital image using one of the first shape and the set ofsecond constituent shapes based on which of the first data point and thesecond data point has a greater respective distance from the hyperplane.20. The system of claim 19, wherein the digital image is segmented usingthe first shape where the where the signed distance between thehyperplane and the first data point is greater than the signed distancebetween the hyperplane and the second data point.
 21. The system ofclaim 20, wherein the digital image is segmented using the set of secondconstituent shapes where the signed distance between the hyperplane andthe second data point is greater than the signed distance between thehyperplane and the first data point.
 22. The system of claim 20, whereinthe memory further includes computer-executable instructions that whenexecuted by the processor cause the processor to apply a watershedtransform to the image data at a predetermined flooding level to producethe set of second constituent shapes.
 23. The system of claim 20,wherein the memory further includes computer-executable instructionsthat when executed by the processor cause the processor to compute alinear combination of each shape in the set of second constituentshapes, wherein the second data point corresponds to the linearcombination.
 24. The system of claim 23: wherein the first shape and thelinear combination of each shape in the set of second constituent shapesare each represented in the image data as a histogram of pointscorresponding to a boundary of the first shape and linear combination,respectively, each point being located on a polar coordinate plane, andwherein the memory further includes computer-executable instructionsthat when executed by the processor cause the processor to compute thefirst data point and the second data point using the histogramcorresponding to the first shape and the linear combination,respectively.
 25. The system of claim 24, wherein the memory furtherincludes computer-executable instructions that when executed by theprocessor cause the processor to rotate each of the first shape and thelinear combination of the set of second constituent shapes such that anaxis of least inertia of the respective shape coincides with a zerodegree radial of the polar coordinate plane prior to computing the firstdata point and the second data point, the axis of least inertiaincluding a line from which the integral of the square of distances toeach point on the boundary of the respective shape is a minimum.